Abstract
Analytical solutions for the growth and vertical and horizontal motion of a precipitation particle growing by coalescence with cloud drops are derived under simplified steady-state assumptions. An equation is also developed for the concentration density of a continuous distribution of growing particles.
Assuming that the cloud water content varies linearly with height, and that the fall speed of a drop is proportional to the square root of its diameter, it is shown that the combination of a linearly increasing updraft surmounted by a sharply decreasing one sets a sharp upper limit to the particle size, and sorts the particles horizontally. Particles which spend their entire life in regions of horizontal convergence associated with increasing updraft are packed into a narrower shaft than that in which they originated. Initially smaller particles are carried above into the region of horizontal divergence associated with decreasing updraft and are displaced far to the sides of the cloud core. It is found that when the updraft increases sharply there is a very small range of initial sizes which can grow to fall-out size. These facts are used to suggest that a steady “balance level” (equal reflectivity in rising and falling particles) may be maintained at a height near and below an updraft maximum. Particle size spectra computed from the concentration density equation are continuous and well-behaved for rising, floating and falling particles alike, without necessarily even maximizing for the floating size.
